# Time:  O(n)
# Space: O(h)

# Given a binary tree, you need to compute the length of the diameter of the tree.
# The diameter of a binary tree is the length of the longest path between
# any two nodes in a tree. This path may or may not pass through the root.
#
# Example:
# Given a binary tree 
#           1
#          / \
#         2   3
#       / \     
#       4   5    
# Return 3, which is the length of the path [4,2,1,3] or [5,2,1,3].
#
# Note: The length of path between two nodes is represented by the number of edges between them.

# Definition for a binary tree node.
# class TreeNode(object):
#     def __init__(self, x):
#         self.val = x
#         self.left = None
#         self.right = None

class Solution(object):
    def diameterOfBinaryTree(self, root):
        """
        :type root: TreeNode
        :rtype: int
        """
        def depth(root, diameter):
            if not root: return 0, diameter
            left, diameter = depth(root.left, diameter)
            right, diameter = depth(root.right, diameter)
            return 1 + max(left, right), max(diameter, 1 + left + right)
            
        return depth(root, 1)[1] - 1
